Persistence theory discussed in this thesis is an application of algebraic topology (Morse Theory) to Data Analysis, precisely to qualitative description of point cloud data. Mathematically a point cloud data is a finite metric space of a very large cardinality. It can be geometrized as a filtration of simplicial complexes and the homology changes of these complexes provide qualitative information about the data. There are new invariants which permit to describe the changes in homology and theseinvariants are the “bar codes”.

In Chapter 3 work is done to develop additional methods for the calculation of bar codes and their refinements. When the coefficient field is Z_2, the calculation of bar codes is done by ELZ algorithm (named after H. Edelsbrunner, D. Letscher, and A. Zomorodian). When the coefficient field is R, we developed an algorithm based on the Hodge decomposition.

The original persistence theory can be viewed as a sort of Morse Theory and involves the “sub level sets” of a nice map. With Dan Burghelea and Tamal Dey we developed a persistence theory about level sets in Chapter 4. This is a refinement of the original persistence. The level persistence is an alternative to Zigzag persistence considered by G. Carlsson and V. D. Silva. I discuss new computable invariantsand how they are related to the known ones. These invariants are referred to as “relevant level persistence numbers” and “positive and negative bar codes”. We provide enhancements and modifications of ELZ algorithm to calculate such invariants and illustrate them by examples.

Chapters 3 and Chapter 4 are preceded by background materials (Chapter 2) where the concepts of algebraic topology used in this paper are defined.